Institute for Mathematical Sciences

Preprint ims12-04

Denis Gaidashev, Tomas Johnson, Marco Martens

Rigidity for infinitely renormalizable area-preserving maps

Abstract: Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown that {\it infinitely renormalizable maps} in a neighborhood of this fixed point admit invariant Cantor sets on which the dynamics is ``stable'' - the Lyapunov exponents vanish on these sets.

Infinite renormalizability exists in several settings in dynamics, for example, in unimodal maps, dissipative H\'enon-like maps, and conservative H\'enon-like maps. All of these types of maps have associated invariant Cantor sets. The unimodal Cantor sets are rigid: the restrictions of the dynamics to the Cantor sets for any two maps are $C^{1+\alpha}$-conjugate. Although, strongly dissipative H\'enon maps can be seen as perturbations of unimodal maps, surprisingly the rigidity breaks down. The Cantor attractors of H\'enon maps with different average Jacobians are not smoothly conjugated. It is conjectured that the average Jacobian determines the rigidity class. This conjecture holds when the Jacobian is identically zero, and in this paper we prove that the conjecture also holds for conservative maps close to the conservative renormalization fixed point.

Rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards
the fixed point. Therefore, to demonstrate rigidity, we prove that the upper bound on the spectral radius of the action of the renormalization derivative on infinitely renormalizable maps is sufficiently small.